Probability Questions

Questions

7E-3) Five people are sitting at a table in a restaurant. Two of them order coffee and the other three order tea. The waiter forgot who ordered what and puts the drinks in a random order for the five persons. Specify an appropriate sample space and determine the probability that each person gets the correct drink.

Answer:

Take an ordered sample space with as outcomes all possible $5!$ orderings of the five people, where the first two people in the ordering get tea from the waiter and the other three get coffee. The number of orderings in which the first two people have ordered coffee and the other people have ordered tea is $2! \times 3!$. Hence the desired probability is $\frac {2! \times 3!} {5!} = 0.1$. Alternatively, we can take an unordered sample space whose outcomes are given by all possible choices of two people from the five people, where these two people get coffee. This leads to the same probability of $1 / {5 \choose 2} = 0.1$

7E-4) A parking lot has 10 parking spaces arranged in a row. There are 7 cars parked. Assume that each car owner has picked at a random a parking place among the spaces available. Specify an appropriate sample space and determine the probability that the three empty places are adjacent to each other.

Answer:

Take an unordered sample space whose outcomes are given by all possible sets of the three different parking places out of the ten parking places. The number of outcomes of the sample space is ${10 \choose 3}$. Each outcome of the sample space gets assigned the same probability of $1/{10 \choose 3}$. The number of outcomes with three adjacent parking places is 8. Hence the desired probability is $8 / {10 \choose 3} = \frac {1} {15}$.

7E-5) Somebody is looking for a top-floor apartment. She hears about two vacant apartments in a building with 7 floors en 8 apartments per floor. What is the probability that there is a vacant apartment on the top floor?

Answer:

Imagine that the 7 x 8 = 56 apartments are numbered as 1, 2, ... , 56. Take as sample space the set of all unordered pairs {i, j} of two distinct numbers from 1, 2, ... , 56, where each pair corresponds to two vacant apartments. The sample space has ${ 56 \choose 2 } = 1,540$ equally likely elements. The number of elements for which there is one apartment vacant on the top floor is ${ 8 \choose 1 } \times { 48 \choose 1 } = 384$ and the number of elements for which there are two apartments vacant on the top floor is ${8 \choose 2} = 28$. Hence the desired probability is $(384 + 28) / 1,540 = 0.2675$. Alternatively, an ordered sampled space can be used to answer the question. Take as ordered sample the set of all 56! permutations of the apartments in the building, where the first two elements in the permutation refer to the vacant apartments. The number of permutations with no vacancy on the top floor is $48 \times 47 \times 54!$. Hence the probability that there is no vacancy on the top floor is $\frac {48 \times 47 \times 54!} {56!} = 0.7325$.

7E-6) You choose at random two cards from a standard deck of 52 cards. What is the probability of getting a ten and hearts?

Answer:

Take as sample space the set of all unordered pairs of two distinct cards. The sample space has ${ 52 \choose 2 } = 1,326$ equally likely elements. There are ${1\choose1} \times {51\choose 1}=51$ elements with the ten of hearts, and ${3\choose1} \times {12\choose1} = 36$ elements with hearts and a ten but not the ten of hearts. Hence the desired probability is $(51 + 36) / 1,326 = 0.0656$.

7E-7) A box contains 7 apples and 5 oranges. The pieces of fruit are taken out of the box, one at a time and in a random order. What is the probability that the bowl will be empty after the last apple is taken from the box?

Answer:

Imagine that the 12 pieces of fruit are numbered as 1, 2, ..., 12. There are 12! possible orders at which the 12 pieces of fruit can be taken out of the box. There are $7 \times 11!$ orders in which the last element is an apple. Hence the probability that the bowl will be empty after the last apple is taken from the box is equal to $\frac {7 \times 11!} { 12!} = 7/12$.

7E-8) A group of five people simultaneously enter an elevator at the ground floor. There are 10 upper floors. The persons choose their exit floors independently of each other. Specify an appropriate sample space and determine the probability that they are all going to different floors when each person randomly chooses one of the 10 floors as the exit floor. How does the answer change when each person chooses with probability 1/2 the 10th floor as the exit floor and the other floors remain equally likely as the exit floor with a probability of 1/18 each.

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7E-9) Three friends and seven other people are randomly seated in a row. Specify an appropriate sample space to answer the following two questions. (a) What is the probability that the three friends will sit next to each other? (b) What is the probability that exactly two of the three friends will sit next to each other?

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7E-10) You and two of your friends are in a group of 10 people. The group is randomly split up into two groups of 5 people each. Specify an appropriate sample space and determine the probability that you and your two friends are in the same group.

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7E-11) You are dealt a hand of four cards from a well-shuffled deck of 52 cards. Specify an appropriate sample space and determine the probability that you receive the four cards J, Q, K, A in any order, with suit irrelevant.

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7E-12) You draw at random five cards from a standard deck of 52 cards. What is the probability that there is an ace among the five cards and a king or queen?

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7E-13) Three balls are randomly dropped into three boxes, where any ball is equally likely to fall into each box. Specify an appropriate sample space and determine the probability that exactly one box will be empty.

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7E-14) An electronic system has four components labeled as 1, 2, 3, and 4. The system has to be used during a given time period. The probability that component i will fail during that time period is $f_i$ for i = 1, . . . , 4. Failures of the components are physically independent of each other. A system failure occurs if component 1 fails or if at least two of the other components fail. Specify an appropriate sample space and determine the probability of a system failure.

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7E-15) The Manhattan distance of a point $(x, y)$ in the plane to the origin $(0, 0)$ is defined as $\|x\|+\|y\|$. You choose at random a point in the unit square ${(x, y) : 0 \leq x, y \leq 1}$. What is the probability that the Manhattan distance of this point to the point $(0, 0)$ is no more than $a$ for $0 \leq a \leq 2$?

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7E-16) You choose at random a point inside a rectangle whose sides have the lengths 2 and 3. What is the probability that the distance of the point to the closest side of the rectangle is no more than a given value $a$ with $0 \lt a \lt 1$?

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7E-17) Pete tosses $n + 1$ fair coins and John tosses $n$ fair coins. What is the probability that Pete gets more heads than John? Answer this question first for the cases $n = 1$ and $n = 2$ before solving the general case.

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7E-18) Bill and Mark take turns picking a ball at random from a bag containing four red balls and seven white balls. The balls are drawn out of the bag without replacement and Mark is the first person to start. What is the probability that Bill is the first person to pick a red ball?

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7E-19) Three desperados A, B and C play Russian roulette in which they take turns pulling the trigger of a six-cylinder revolver loaded with one bullet. Each time the magazine is spun to randomly select a new cylinder to fire as long the deadly shot has not fallen. The desperados shoot according to the order A,B,C,A,B,C,... . Determine for each of the three desperados the probability that this desperado will be the one to shoot himself dead.

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7E-20) A fair coin is tossed 20 times. The probability of getting the three or more heads in a row is 0.7870 and the probability of getting three or more heads in a row or three or more tails in a row is 0.9791. What is the probability of getting three or more heads in a row and three or more tails in a row?

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7E-21) The probability that a visit to a particular car dealer results in neither buying a second-hand car nor a Japanese car is 55%. Of those coming to the dealer, 25% buy a second-hand car and 30% buy a Japanese car. What is the probability that a visit leads to buying a second-hand Japanese car?

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7E-22) A fair die is repeatedly rolled and accumulating counts of 1s, 2s, ... , 6s are recorded. What is an upper bound for the probability that the six accumulating counts will ever be equal?

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7E-23) A fair die is rolled six times. What is the probability that the largest number rolled is r for r = 1, ... , 6?

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7E-24) Mr. Fermat and Mr. Pascal are playing a game of chance in a cafe in Paris. The first to win a total of ten games is the overall winner. Each of the two players has the same probability of 1/2 to win any given game. Suddenly the competition is interrupted and must be ended. This happens at a moment that Fermat has won $a$ games and Pascal has won $b$ games with $a \lt 10$ and $b \lt 10$. What is the probability that Fermat would have been the overall winner when the competition would not have been interrupted? Hint : imagine that another $10 - a + 10 - b - 1$ games would have been played.

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7E-25) A random number is repeatedly drawn from 1, 2, ... , 10. What is the probability that not all of the numbers 1, 2, ... , 10 show up in 50 drawings?

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7E-26) Three couples attend a dinner. Each of the six people chooses randomly a seat at a round table. What is the probability that no couple sits together?

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7E-27) You roll a fair die six times. What is the probability that three of the six possible outcomes do not show up and each of the other three possible outcomes shows up two times? What is the probability that some outcome shows up at least three times?

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7E-28) In a group of $n$ boys and $n$ girls, each boy chooses at random a girl and each girl chooses at random a boy. The choices of the boys and girls are independent of each other. If a boy and a girl have chosen each other, they form a couple. What is the probability that no couple will be formed?

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7E-29) Twelve married couples participate in a tournament. The group of 24 people is randomly split into eight teams of three people each, where all possible splits are equally likely. What is the probability that none of the teams has a married couple?

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7E-30) An airport bus deposits 25 passengers at 7 stops. Each passenger is as likely to get off at any stop as at any other, and the passengers act independently of one another. The bus makes a stop only if someone wants to get off. What is the probability that somebody gets off at each stop?

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7E-31) Consider a communication network with four nodes $n\_1, n\_2, n\_3$ and $n\_4$ and five directed links $l_1 = (n_1, n_2), l_2 = (n_1, n_3), l_3 = (n_2, n_3), l_4 = (n_3, n_2), l_5 = (n_2, n_4)$ and $l_6 = (n_3, n_4)$. A message has to be sent from the source node n1 to the destination node n4. The network is unreliable. The probability that the link $l_i$ is functioning is pi for i = 1, . . . , 5. The links behave physically independent of each other. A path from node n1 to node n4 is only functioning if each of its links is functioning. Use the inclusion-exclusion formula to find the probability that there is some functioning path from node n1 to node n4. How does the expression for this probability simplify when pi = p for all i?

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Combinatorial Problems

3.1.1) If in a survey of 100 people, 65 people drink, 28 smoke, and 30 do neither, then how many do both?

Answer: 23

Conditional Probability

1.5) A family has two children. What is the conditional probability that both are boys given that at least one of them is a boy? Assume that the sample space S is given by S = {(b, b), (b, g), (g, b), (g, g)}, and all outcomes are equally likely. ((b, g) means, for instance, that the older child is a boy and the younger child a girl.)

Answer: $\frac {1/4} {3/4} = \frac {1} {3}$

1.6) Bev can either take a course in computers or in chemistry. If Bev takes the computer course, then she will receive an A grade with probability 1/2; if she takes the chemistry course then she will receive an A grade with probability 1/3. Bev decides to base her decision on the flip of a fair coin. What is the probability that Bev will get an A in chemistry?

Answer: $\frac {1} {2} . \frac {1} {3} = \frac {1} {6}$

1.7) Suppose an urn contains 7 black balls and 5 white balls. We draw 2 balls from the urn without replacement. Assuming that each ball in the urn is equally likely to be drawn, what is the probability that both drawn balls are black?

Answer: $\frac {7} {12} . \frac {6} {11} = \frac {42} {132}$

1.8) Suppose that each of three men at a party throws his hat into the center of the room. The hats are first mixed up and then each man randomly selects a hat. What is the probability that none of the three men selects his own hat?

Answer:

Probability that each man selects his own hat is, $\frac {1} {6}$. At least one man selects his own hat, $P(E_1 \cup E_2 \cup E_3) = P(E_1) + P(E_2) + P(E_3) - P(E_1 E_2) - P(E_1 E_3) - P(E_2 E_3) + P(E_1 E_2 E_3) = 1 - \frac {1} {2} + \frac {1} {6} = \frac {2} {3}$None of them selects his own hat, $1 - \frac {2} {3} = \frac {1} {3}$